A neon sign spells out Azealia Banks's name atop a bowl of noodles, in the style of a New York Chinese takeaway. When the latest US hip-hop sensation walks onstage, detonating a confetti cannon, her slight frame is wrapped in a blue Chinese dress and strafed by hair extensions worthy of Rapunzel. As yet, she is probably bigger over here than she is over there. Banks has just finished opening NME's new bands tour; this sold-out show is Banks's second headline slot in the capital since last autumn, when her track "212" went viral. Much is expected of this remarkable Harlem rapper. Recently signed to Universal, she has played a gig for Karl Lagerfeld and announced a debut album for the autumn. Entitled Broke With Expensive Tastes, Banks has been working on it with the help of Paul Epworth (Adele, Florence Welch) and, Twitter has it, MIA.

Tonight, though, that album seems both promising and still in development. The sound quality is patchy, and her half-hour set seems more "hip niche" than it does a bid for mainstream stardom. But there is no denying Azealia Banks's appeal. Like Nicki Minaj, Banks has swiftly become infamous for having a mouth so filthy that she probably needs an industrial pressure hose to brush her teeth. You suspect many people are here to shout out one gloriously provocative line in "212", "I guess that cunt gettin' eaten!"

This brand of fierce-girl filth, all penile put-downs and catty call-outs, is nothing new. It has been the template for most female rappers passim bar Missy Elliott, who managed to squirrel a little surreality into the game. Somehow, though, the smiley, mischievous, stage-schooled Banks has brought a freshness to these motions, like a bratty little sister who is smarter, smuttier and more versatile than you'd feared. Banks's flow is fast and thick with complex riffs on assonance; she can also whip out a cover of "Valerie", the Zutons song made famous by Amy Winehouse. Where Minaj has pioneered a cartoon hyper-sexuality, tonight Banks is channelling a kitsch chic that's every bit as metropolitan and internationalist as it is urban.

Banks's DJ, Cosmo, is from the electronic music hub of Montreal and the beats bubbling around her verbiage hail from all corners of the digital map. After a furious opening a cappella called "Grandslam" in which Banks mocks men with language-bending vowel flurries, "Fuck Up the Fun" finds her machine-gunning to hollow slapping rhythms that summon up both Brazil and Baltimore. Later, "Bambi" proffers a less successful ravey European nightclub fusion. "Licorice" – an inter-racial come-hither with a sordidly sweet tooth – is stark house music with a nagging sung hook. It signposts one possible path forward into commerce.

New Jersey's Sharon Van Etten hasn't had anywhere near as much attention as Banks. Despite her relative lack of dick metaphors, however, this singer-songwriter has made an equally assured start to 2012. Sex does figure – "It might be I always give out," Van Etten sighs at the end of "Give Out" – but finely wrought dissections of emotional states dominate.

Van Etten's star has risen more surreptitiously than Banks's, with the help of Bon Iver (who covered one of her old songs) and the National's Aaron Dessner, a Brooklyn indie-rock enabler of some note who produced her most recent album. Emotionally saturated and dense with circular patterns of guitar, Tramp has accrued rave reviews. On the first night of her UK tour, its heady thrum is recreated with the help of an omnichord and a full band who swap instruments and self-deprecating anecdotes. Backing vocalist Heather Woods Broderick is pivotal, adding more velvet to Van Etten's gliding vocals.

Confessional indie rock can be dull, po-faced and samey, but tonight Van Etten and her band display a hypnotic, off-kilter shimmer like that which Throwing Muses once brought to the alternative rock of the late 80s. And "Serpents", their most seething three minutes, finds Van Etten at her most aggressive and forthright: "You enjoy sucking on dreams," she sneers, "so I will fall asleep with someone other than you."